Symmetry reduction, integrability and reconstruction in k-symplectic field theory

Abstract

We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a k-connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' k-connection.

37 pages

References in corpus (7)

• Lagrange-Poincare field equations
• Reduction of polysymplectic manifolds
• Euler-Poincare reduction for discrete field theories
• Reduction and reconstruction aspects of second-order dynamical systems with symmetry
• Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations
• Symmetries, Newtonoids vector fields and conservation laws in the Lagrangian $k$-symplectic formalism
• SOPDEs and Nonlinear Connections

Cited by in corpus (3)

• Conditions for symmetry reduction of polysymplectic and polycosymplectic structures
• Multisymplectic observable reduction using constraint triples
• Cotangent bundle reduction and Routh reduction for polysymplectic manifolds